To link to this article : DOI:10.1017/jfm.2012.288 URL : http://dx.doi.org/10.1017/jfm.2012.288
To cite this version : Bonhomme, Romain and Magnaudet, Jacques and Duval, Fabien and Piar, Bruno Inertial dynamics of air bubbles crossing a horizontal fluid–fluid interface. (2012) Journal of Fluid Mechanics, vol. 707. pp. 405-443. ISSN 0022-1120
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1
Inertial dynamics of air bubbles crossing a
horizontal fluid–fluid interface
Romain Bonhomme1,2,3, Jacques Magnaudet1,2†, Fabien Duval3
and Bruno Piar3
1Universit´e de Toulouse; INPT, UPS; IMFT (Institut de M´ecanique des Fluides de Toulouse);
All´ee Camille Soula, F-31400 Toulouse, France
2CNRS; IMFT; F-31400 Toulouse, France
3Institut de Radioprotection et de Sˆuret´e Nucl´eaire, BP 3, 13115 St Paul lez Durance CEDEX, France
The dynamics of isolated air bubbles crossing the horizontal interface separating two Newtonian immiscible liquids initially at rest are studied both experimentally and computationally. High-speed video imaging is used to obtain a detailed evolution of the various interfaces involved in the system. The size of the bubbles and the viscosity contrast between the two liquids are varied by more than one and four orders of magnitude, respectively, making it possible to obtain bubble shapes ranging from spherical to toroidal. A variety of flow regimes is observed, including that of small bubbles remaining trapped at the fluid–fluid interface in a film-drainage configuration. In most cases, the bubble succeeds in crossing the interface without being stopped near its undisturbed position and, during a certain period of time, tows a significant column of lower fluid which sometimes exhibits a complex dynamics as it lengthens in the upper fluid. Direct numerical simulations of several selected experimental situations are performed with a code employing a volume-of-fluid type formulation of the incompressible Navier–Stokes equations. Comparisons between experimental and numerical results confirm the reliability of the computational approach in most situations but also points out the need for improvements to capture some subtle but important physical processes, most notably those related to film drainage. Influence of the physical parameters highlighted by experiments and computations, especially that of the density and viscosity contrasts between the two fluids and of the various interfacial tensions, is discussed and analysed in the light of simple models and available theories.
Key words: capillary flows, drops and bubbles, multiphase flows
1. Introduction
Buoyancy-driven drops and bubbles crossing horizontal liquid–liquid interfaces are encountered in a variety of engineering situations such as liquid–liquid extraction, emulsification or iron processing. For instance, the ladle stirring technique widely employed in iron processing makes use of nitrogen bubbles injected at the bottom of the device to stir and mix the liquid metal and remove impurities (Poggi, Minto & Davenport 1969; Kobayashi 1993). Some nuclear accident scenarios also consider the
situation where the concrete slab below the reactor is ablated by the fuel-containing material (corium) and releases gas bubbles. In both cases, due to compositional differences, the fluid has a two-layer structure with an upper layer made of slag in the former case and predominantly of oxides in the latter. Similar configurations are found in geophysical problems where fluids undergo natural discontinuous density and/or viscosity stratification, such as the ascent of plumes through the Earth’s mantle (Manga, Stone & O’Connell 1993). In microfluidics, two-layer fluid systems may be used to coat paramagnetic drops or particles with a shell made of the lower fluid, buoyancy then being replaced by a magnetic force (Tsai et al. 2011).
Most early experimental and theoretical investigations of that problem were motivated by the fact that it may be considered as a canonical situation for understanding coalescence, especially the various stages of the drainage of the film located between the top part of the bubble and the fluid–fluid interface (Charles & Mason 1960; Allan, Charles & Mason 1961; Princen 1963; Princen & Mason
1965a,b). Hence they usually focused on creeping flow conditions and frequently considered the particular situation where the drop or bubble is made of the same fluid as one of the other two phases (see Chi & Leal 1989 and Mohamed-Kassim & Longmire 2004 for reviews). Then the problem began to be considered numerically, either in the limit of creeping motion using boundary integral techniques (Chi & Leal 1989; Manga & Stone 1995), or in presence of finite inertial effects by solving the full Navier–Stokes equations on a boundary-fitted grid (Shopov & Minev 1992). Nevertheless only the early stages of the motion during which the drop approaches the fluid–fluid interface were considered in these computations because they could not deal with film break-up, nor with the subsequent topological changes of the flow. In the recent period, the development of high-speed video imaging techniques and particle image velocimetry has allowed the case of high-Reynolds-number gas bubbles crossing fluid–fluid interfaces to be investigated in more detail (Reiter & Schwerdtfeger 1992a; Kemiha et al. 2007; Dietrich et al. 2008). In particular the evolution of the bubble shape and that of the column of heavy liquid it entrains under certain conditions were characterized in several regimes.
A closely related configuration that has received a great deal of attention is that of a rigid sphere approaching or crossing a horizontal fluid–fluid interface. Again this problem was initially considered in view of its connection with coalescence, and most investigations focused on the film-drainage configuration in which the sphere rests very close to an interface which only weakly deforms (Hartland 1968, 1969; Shah, Wasan & Kintner 1972; Jones & Wilson 1978; Smith & Van de Ven 1984). Nevertheless some investigations rather considered, either experimentally (Maru, Wasan & Kintner
1971) or numerically (Leal & Lee 1982; Geller, Lee & Leal 1986), situations in which the interface deformation may become large as time proceeds, the sphere then towing a long column or tail of heavy fluid with it. The case of heavy spheres settling across a sharp density interface separating two miscible fluids has also been considered experimentally, especially in connection with its relevance to the prediction of pollutant dispersion (Srdi´c-Mitrovi´c, Mohamed & Fernando 1999; Camassa et al.
2009). Finally it is worth mentioning that pseudo-three-phase systems with the same geometry have been used to study the entrainment of fluid or rigid particles by rising bubbles, especially in fluidized beds. In this case the two fluid layers are made of a single liquid and entrainment of the lower layer is quantified by marking it, for instance with dye or milk, and evaluating the volume of the displaced fluid. When the flow disturbance is close to that predicted by potential flow theory, this displaced
volume may valuably be related to the classical concept of Darwin’s drift (Eames & Duursma1997; Bush & Eames 1998).
The present paper reports on a joint experimental and computational study of the above problem in the case of a single gas bubble crossing an interface between a lower phase made of water or water plus glycerin and an upper, slightly lighter, phase made of silicon oil. Varying the size of the bubble, the glycerin concentration and the characteristics of the silicon oil allows us to explore a broad range of physical conditions, leading to a variety of flow configurations and bubble shapes. The main purpose of this investigation is to obtain new insight into the key features of particular interest in this problem, namely the influence of film drainage on the bubble dynamics when it reaches the fluid–fluid interface, the final topology of the three phases, and, in cases where the bubble succeeds in crossing the interface, the evolution of its rise speed and of the volume of heavy fluid it entrains during its ascent in the upper fluid. Experimental data are obtained by means of high-speed video imaging and in some cases particle image velocimetry. Computations are based on the so-called volume of fluid approach in which the incompressible Navier–Stokes equations are solved on a fixed grid. As will be seen, the two approaches efficiently complement each other. For instance computations help reveal important details of the flow that cannot be obtained with the present optical technique, owing to limitations in its spatial resolution. They also give access to flow regions that may not be reached optically, due to the bubble shape (e.g. with toroidal or spherical caps bubbles with a concave base). Last but not least they guarantee that the various interfaces are surfactant-free and allow each physical property of the fluids to be varied independently for all others. However, three-phase flows involve complex small-scale phenomena such as film drainage, break-up and moving contact lines. Some of these phenomena may not be properly captured in computations, owing to the spatial cut-off introduced by the computational grid and to the approximate representation of interfacial forces. This is why direct computations of such flows cannot yet be performed in a blind manner and their results have to be compared with those of experiments to make sure that the former correctly predict the global dynamics revealed by the latter.
The paper is organized as follows. Sections 2 and 3 describe the experimental and computational approaches, respectively; specific technical details on the latter and validation tests are provided in three separate appendices. Section4 presents an overview of the physical behaviours revealed by the entire set of experiments. In §5, we select six situations corresponding to contrasting flow conditions and bubble shapes and analyse each of them in detail with the help of experimental observations and computational predictions. Section6 relies on some of the experimental and computational results and on a simple static model developed in appendixDto analyse the elementary mechanisms that drive the evolution of the system when the bubble reaches the fluid–fluid interface. Finally, §7 summarizes the main findings obtained during this investigation and opens up some perspectives.
2. Experimental device and measurement techniques
Experiments are carried out at ambient temperature (20±1◦C) in a glass tank 40 cm
high with a 20 cm × 20 cm square cross-section. Two sides of the tank are made of B270 Superwite R glass to limit optical distortions. The upper part of each vertical wall
is treated with a Rain-X R hydrophobic compound to avoid meniscus effects along the
glass/water/oil contact line. The base of the tank is made of Plexiglas (PMMA); it is removable and comprises several interchangeable injection systems. After each series
L. F. generator Computer Telecentric lens CCD camera Syringe Drain Water tank M
FIGURE1. (Colour online) Sketch of the experimental device with the inverted beaker injection system. L.F., low frequency.
of measurements, each liquid is removed from the tank with a siphon at least every two days and stored in closed containers. The residual liquid layers adjacent to the liquid–liquid interface are thrown out. The tank is then washed with a detergent liquid, rinsed out with tap water a large number of times, and dried with a duster.
We basically employ two different gas injection techniques. One of them (shown schematically in figure 1) is inspired by the ‘inverted beaker’ used by Davies & Taylor (1950) to produce large spherical cap bubbles. This device is made of a ‘spoon’ 60 mm in diameter, which may be rotated by hand about a horizontal axis.
A controlled air volume is injected below the spoon through a syringe. Then the spoon is turned over to release the bubble. This device is suitable for generating spheroidal and spherical cap bubbles. In the second injection system, the air volume is initially entrapped in a closed cylinder and then rapidly injected manually in the tank by pushing a piston. This device is mostly used with large air volumes (typically >1 cm3) to obtain toroidal bubbles. Based on the comparison between the injected
volume of air and the optical determination (to be described below) of the bubble contour for small spherical bubbles, the uncertainty in the bubble equivalent diameter d = (6V /π)1/3, where V is the bubble volume, was found to be 50 µm, so that the relative uncertainty in d for bubbles injected with the first system ranges from 3 % for the smallest of them (d ≈ 1.5 mm) to less than 0.5 % for the biggest (d ≈ 20 mm).
Two high-speed digital Photon Lines PCO1200 HS cameras with a resolution of 1024 × 1280 pixels synchronized at a rate of 350 images per second are employed to visualize the bubble and the evolution of the various fluid–fluid interfaces. The two cameras are placed at right angles perpendicular to two of the vertical glass walls of the tank. We use backlighting to obtain the projections of the bubble and fluid–fluid interface shadow in the visualization planes (the schlieren technique). In
order to reduce optical distortions, one of the cameras is equipped with an Opto Engineering telecentric lens TC 4M 120. The system is calibrated in such a way that each camera detects a fixed field of 108.9 mm × 87.2 mm. Contours of bubbles, droplets and columns of heavy liquid displaced through the upper fluid are detected on the images with a thresholding method followed by an erosion–dilatation process. Positions and surface elements identified on a given frame are then tracked in time using a maximum likelihood detection process. In some cases we also employ particle image velocimetry (PIV) to determine the velocity field past the bubble. For that purpose, a 2 × 25 mJ Pegasus laser source lights up the two fluids seeded by Rhodamine-B molecules encapsulated in PMMA particles whose size is in the range 1–20 µm. The velocity fields are then extracted from intercorrelations of (16×16)-pixel elements using the in-house software PIVIS.
In the experiments to be described below, the lower (and thus heavier) liquid is either tap water or a mixture of glycerin and tap water with two different volume fractions of glycerin, 85 or 95 %. No particular treatment is applied to water, so that surfactants are likely to be present and one may suspect them to somewhat lower the rise velocity of small spherical or spheroidal bubbles compared to theoretical predictions assuming a clean bubble surface; with usual surfactant concentrations, larger bubbles (say with d > 4 mm) are known to be much less influenced by contamination. The upper (and thus lighter) liquid is silicon oil. Three different oils (47V10, 47V100 and 47V500 from Gaches Chimie company) are employed, with viscosities approximately ranging from 10 to 500 times that of water. The physical properties of all fluids were determined at room temperature (20 ± 1◦C).
Viscosity and surface tension were obtained using a Bohlin cone-plate viscometer and a Wilhelmy plate device, respectively, while a drop shape analysis system, Kruss DSA100, based on the pendant-drop method was employed to measure the interfacial oil/water + glycerin tensions. The relative uncertainty in the viscosity is 5 % for the least viscous oil and is a decreasing function of viscosity. The uncertainty in surface and interfacial tensions is 1.5 mN m−1, so that the corresponding relative
uncertainty ranges from 2 (for the water/air system) to 11 % (for the water/47V500 oil system). Density was determined by weighing a 100 ml calibrated flask filled with the corresponding liquid on a precision balance with a 0.1 g accuracy, so that the corresponding relative uncertainty is ∼0.1 %. The various physical properties relevant to the systems described below are summarized in table1.
3. Computational approach
In the context of the one-fluid approach, a three-phase flow is considered as a mixture of three immiscible fluids. It may be characterized by the local volume fraction Ci (i =1, 2, 3) of each of them and by density and viscosity fields that
depend only on Ci and on the corresponding intrinsic physical property of each
fluid. Assuming all fluids to be Newtonian and the various interfaces to be sharp with uniform interfacial tensions, the system of equations governing the motion of an incompressible three-phase flow is then given by
DCi Dt =0 for i = 1, 2, 3, (3.1) ∇ · U = 0, (3.2) ̺DU Dt = ̺g − ∇P + ∇ · {µ(∇U + ∇U T )} + T , (3.3)
Liquid Density
(kg m−3) Viscosity(mPa s) Surface tension(mN m−1)
Water 997 1.002 69.4 85 % glycerin + water 1211 102.6 48.7 95 % glycerin + water 1244 550.1 45.2 47V10 silicon oil 932 9.6 20.2 47V100 silicon oil 961 113.8 20.7 47V500 silicon oil 965 530.7 21.0
Interfacial tension (mN m−1) 47V10 oil 47V100 oil 47V500 oil
Water 19.7 14.3 13.7
85 % glycerin + water 28.8 30.0 29.9
95 % glycerin + water 27.8 27.1 28.0
TABLE 1. Physical properties of the various liquids measured at 20 ± 1◦C.
where D/Dt = ∂/∂t + U · ∇ denotes the material derivative, T is the capillary force density, and the density and viscosity of the mixture are related to the volume fraction of each phase through the linear laws
̺ = 3 X i=1 Ci̺i, µ = 3 X i=1 Ciµi. (3.4)
Note that the first of (3.4) is an exact result while the second is just an ad hoc interpolation formula. The JADIM code developed at IMFT solves the above set of equations with a capillary force density defined as
T= −1 2 3 X i=1 3 X j=1 (Ci+ Cj)2σij∇ · ∇Cij k∇Cijk ∇Cij, (3.5)
where Cij= Ci/(Ci+ Cj) =1 − Cji. The derivation of (3.5), which is an extension of
the continuum surface force (CSF) formulation (Brackbill, Kothe & Zemach 1992), is detailed in appendix A; comparison with alternative formulations is also discussed there.
The code is based on a finite-volume discretization combined with a
Runge–Kutta/Crank–Nicolson time-advancement scheme; incompressibility is enforced via a projection method. Centred schemes are used to approach the various spatial derivatives involved in (3.3) while the solution of (3.1) is based on a direction-splitted version of Zalesak’s Flux-Corrected Transport algorithm (Zalesak 1979). The full numerical approach has been extensively described by Bonometti & Magnaudet (2007) (see also Bonometti & Magnaudet 2006) and will not be repeated here. However, the way we deal with the issue of volume conservation of each phase in a three-phase system deserves some specific comments, which are provided in appendix B.
Throughout the paper, axisymmetric computations are performed within a vertical cylindrical domain with a 6d radius and a 12d height, the initial fluid–fluid interface standing 7d from the top of the domain. The grid employs 1200 cells uniformly distributed in the vertical direction. In the horizontal direction, 200 grid points are uniformly distributed in the central region extending up to 2d from the axis, while another 100 points are distributed in the outer region following an arithmetic law. The present resolution, with a hundred cells per bubble diameter, was used by Bonometti &
Magnaudet (2006, 2007) who found it to provide grid-independent results for various types of bubbles rising in a Newtonian fluid, including toroidal bubbles; this is why it is selected again here. Free-slip boundary conditions are imposed on the top, bottom and lateral boundaries, so that the fluid entrained upwards by the bubble slowly goes down near the lateral boundary. Computations are stopped before the bubble gets close to the upper boundary, to avoid contamination of results by confinement effects.
The code was extensively validated in the past in various two-phase configurations. Simple three-phase configurations for which a theoretical solution exists, such as the spreading of a small lens at the interface between two liquids (de Gennes, Brochard-Wyart & Qu´er´e 2004), were also considered. These tests showed that the theoretical shape of the lens is recovered under various conditions of spreading, which validates the above formulation for the capillary force T . An additional test case in a physical situation close to those under focus here and employing the grid characteristics described above is detailed in appendix C. This test deals with the early evolution of low-Reynolds-number buoyancy-driven drops of various viscosities rising toward a horizontal interface separating two fluids with different viscosities and densities, a configuration that was computed by Manga & Stone (1995) using a boundary integral method (BIM). The results of both approaches turn out to be in excellent agreement until the drop gets very close to the interface between the lower and upper fluids. More precisely, with low-viscosity drops behaving very similarly to bubbles, differences from the predictions of Manga and Stone start to be significant when the thickness of the film separating the top of the drop from the upper fluid becomes .0.04d. These differences result from two phenomena, both of which are related to the finite thickness of the interfaces in the present approach. First, the capillary force density (3.5) involves the gradients of Ci and then spreads out over several cells across
an interface. Second, the interpolation law (3.4) used to determine the local viscosity implies that, within an interfacial region, viscosity takes values intermediate between those of the two fluids in contact. Both phenomena become increasingly important within the film as it thins: being bounded by two interfaces, the flow in the gap experiences some artificial capillary force and the viscosity is no longer that of the actual fluid. The consequences of these artifacts will be seen in more detail in §5.
4. Overview of experimental observations
We start by providing a qualitative survey of the evolution of the three-phase system as its characteristic parameters are varied. For this purpose, a prerequisite is the definition of a proper set of dimensionless characteristic numbers. General three-phase systems involving Newtonian fluids are characterized by nine physical properties, namely three densities (ρi, i =1, 2, 3), three viscosities (µi) and three
interfacial tensions (σij with j 6= i and σij= σji). Moreover the dynamical problem
depends on gravity g and assuming that the bubble is initially spherical and located far from the liquid–liquid interface within a flow domain extending up to infinity, on a single length scale, for instance the bubble equivalent diameter d. As these eleven quantities involve three fundamental units (mass, time and length), the problem may be characterized with eight independent dimensionless parameters. However, since the bubble viscosity and density are negligibly small compared to those of the various liquids we use, they may be removed from the list of relevant quantities, which reduces the number of independent dimensionless parameters to six. Let fluid 1 (respectively 3) refer to the lower (respectively upper) liquid as in the previous section. Then, defining all parameters with respect to the properties of
fluid 1, we may select R = (ρ1− ρ3)/ρ1, Λ = µ3/µ1, I = σ13/σ12, S = σ23/σ12, Bo =
ρ1gd2/σ12, Ar = ρ1(gd3) 1/2
/µ1. The Bond (or E¨otv¨os) number Bo compares buoyancy
effects to capillary effects while the Archimedes number Ar may be thought of as a Reynolds number based on the gravitational velocity (gd)1/2. The problem is then entirely defined by the set of parameters (R, Λ, I, S, Bo, Ar).
Given the values of the physical properties reported in table 1, the viscosity ratio Λ varies by more than four orders of magnitude through the whole set of experiments, from 0.0175 (when the lower liquid is a 95 % glycerin + water mixture and the upper liquid is the 47V10 silicon oil) to 530 with the water/47V500 silicon oil system. The density ratio R varies from 0.036 to 0.251 (note that, owing to its definition, R always satisfies the constraint 06 R 6 1). The parameter I, a dimensionless measure of the strength of interfacial effects at the fluid–fluid interface, varies by a factor of three between series A or B and series D (0.216 I 6 0.62), while the ratio S of surface tensions in the upper and lower fluids experiences a more modest variation (0.296 S 6 0.465). In view of future discussions, it is also useful to introduce the Archimedes and Bond numbers in the upper fluid, Aru and Bou, defined as
Aru= (1 − R)Ar/Λ and Bou= (1 − R)Bo/S, respectively.
Images of some of the bubbles emerging in the upper fluid and of the associated entrained volume of heavy fluid (if any) are displayed in figure 2. Each two-dimensional diagram in that figure corresponds to a given set of fluids (i.e. given values of R, Λ, I and S); the Bond and Archimedes numbers in each series are varied by generating bubbles of increasing volume, i.e. increasing d. As suggested by figure 2, small enough bubbles, typically those with Bo < 5 in series A or Bo <7 in series B, remain trapped at the interface between the two liquids during the entire period of observation (typically 1 mn in this range of Bo). That is, no macroscopic change has occurred in the system at the time we stop recording images but this obviously does not rule out the possibility of later changes, since these bubbles are expected to be covered by a very thin film of heavy liquid that should eventually be drained, leading to the release of the bubble after the film has ruptured. This entrapment corresponds to situations in which capillary effects result in a downward force capable of balancing the upward buoyancy force. Although the full film-drainage problem has been worked out in great detail in the low-Reynolds-number limit (Princen 1963; Princen & Mason 1965a,b), it does not have a general tractable theoretical solution, even in that limit. A crude criterion predicting conditions under which bubble entrapment occurs was derived by Greene, Chen & Conlin (1988). Based on the idea that a bubble cannot be stopped at the interface if the buoyancy force it experiences in the light fluid exceeds the maximum possible interfacial capillary force (reached when the contact line lies in the horizontal mid-plane of the bubble and the meniscus is vertical), this criterion is in the present notation given by Bo <6I/(1 − R) if the bubble is assumed to be spherical. It suggests that bubbles of series A and B should not be stopped near the undisturbed position of the interface for Bo > 5 approximately, while those in series C (respectively D) should not be stopped for Bo > 1.8 (respectively 1.3). According to figure 2, this criterion predicts the trapped/untrapped transition reasonably well. We shall consider the mechanisms underlying this transition in more detail in §6.
Bubbles that are not trapped near the interface then rise in the upper liquid and tow a column (or tail) of heavy liquid for some time. Experimental observations suggest that after the bubble has travelled a distance of a few d above the initial position of the interface, this column is most often directly attached to the rear part of the bubble, i.e. no film is discerned on the front part of the bubble at this late stage,
101 100 100 101 102 Ar Ar Series A Series C Series D Upper series B Lower series B 100 101 102 100 101 102 101 101 104 103 102 104 103 102 Bo 100 102 100 102 Bo (a) (c) (d ) (b)
FIGURE 2. Some selected shapes of bubbles and entrained columns of heavy fluid in five series of experiments (only the bubble size is varied in each series). (a) Series A (95 % glycerin + water/47V10 oil, 36 Bo 6 50, 1.5 6 Ar 6 12, i.e. 5 6 Bou6 83, 64 6 Aru6 514). (b) Lower series B (95 % glycerin + water/47V500 oil, 3.86 Bo 6 55, 1.3 6 Ar 6 12, i.e. 6.36 Bou6 92, 1.0 6 Aru6 9.6); upper series B (85 % glycerin + water/47V100 oil, 2.1 6 Bo6 50, 7.0 6 Ar 6 60, i.e. 3.9 6 Bou6 93, 5.0 6 Aru6 43). (c) Series C (water/47V10 oil, 1.26 Bo 6 50, 550 6 Ar 6 8.0 × 103, i.e. 3.96 Bou6 160, 54 6 Aru6 780). (d) Series D (water/47V100 oil, 1.36 Bo 6 52, 500 6 Ar 6 7.4 × 103, i.e. 4.26 Bou6 167, 1.6 × 1036 Aru 6 23.8 × 103). In each image, the vertical axis is positioned on the corresponding value of Bo while the horizontal trace of the flat fluid–fluid interface is positioned on the corresponding value of Ar.
implying that film drainage has already been completed and that a moving contact line subsequently develops between the bubble and the two liquids. A noticeable exception is that of toroidal bubbles, which turn out to remain completely encapsulated in a thick film of heavy liquid. The column may sometimes break up into droplets when Λ ≫ 1 (e.g. Bo ≈ 5 and Ar ≈ 1.3 × 103 in series D) because it is then strongly sheared by the
upper fluid that hardly deforms. In the opposite case where the upper liquid deforms much more easily than the lower one (Λ ≪ 1), bubbles are accelerated after they have crossed the interface (although the buoyancy force is somewhat less than in the lower fluid). This acceleration may totally suppress the entrained column because it may break the bubble when it is close to emerging entirely from the interface, forcing its former rear part to remain trapped between the two liquids (e.g. Bo ≈ 9 and Ar ≈ 3.2 in series A).
Depending on their final Bond and Reynolds numbers, rising bubbles may look like oblate spheroids (left part of series C and D or central part of series B), exhibit a marked tail (central part of series A or right part of the lower series B), look like spherical caps (right part of series A and of the upper series B, central part of series C and D) or, for large enough Bond and Archimedes numbers, adopt a toroidal shape (upper right corner of series C and D). According to the position at which bubbles approximately switch from one shape to the other, the transition from spheroidal to spherical cap bubbles may be estimated to take place at Bou≈20
provided Aru> 20, while that from spherical cap to toroidal bubbles takes place
at Bou ≈125 and Aru ≈50. As it is directly related to the width of the wake,
the head of the entrained column is significantly thicker with spherical caps than with spheroidal bubbles, resulting in a larger displaced volume. Also, for a given Bond number and position above the undisturbed interface, the larger the Archimedes number (and thus the Reynolds number) in the upper fluid, the thinner the wake and hence the smaller the displaced volume (compare the two snapshots in the upper right corner of series A and B, where Aru is ∼12 times larger in the former series).
Most configurations displayed in figure 2 are axisymmetric. Fully three-dimensional configurations are only observed in series C and D when Ar > 103. In this regime,
which also corresponds to O(103) Reynolds numbers, the wake of axisymmetric bluff
bodies is known to be three-dimensional, whether they are spheres or disks obeying a no-slip condition (Natarajan & Acrivos 1993) or stress-free spheroidal bubbles with a sufficient oblateness (Magnaudet & Mougin 2007). When the corresponding bubbles rise in the upper fluid, their actual Reynolds number drops by one (respectively two) order of magnitude in series C (respectively D) but in most cases this is still sufficient for the wake to remain unstable, resulting in tortuous bubble shapes and columns of entrained fluid. Remarkably, for a given Bond number, the bubble and displaced volume geometries in series C and D are quite similar as soon as Bo exceeds some units, although the actual Archimedes number in the upper liquid differs by one order of magnitude between the two series. This is an indication that the flow about the bubble is only weakly affected by viscous effects in these strongly inertial regimes.
5. Analysis of experimental and computational results in some selected situations
To explore the dynamics of the three-phase system in more detail, we select some situations within figure 2. One of them is typical of small bubbles (with Bo = O(1)) that remain trapped at the interface throughout the period of observation. The following two are characterized by moderate, say O(10), Bond numbers and result in bubbles with a more or less spheroidal shape, possibly with some cusp at the rear. They differ in the strength of inertial effects, which are weak in one case while they dominate in the other. Two other situations belong to regimes characterized by Bond numbers of several tens where the bubble eventually takes a spherical cap shape; the magnitude of inertial effects in the upper fluid differs by one order of magnitude
(a)
(b)
FIGURE 3. (Colour online) Evolution of the three-phase system for a bubble corresponding to Bo = 3.0 and Ar = 7.65 in the upper series B. Comparison between (a) the experimental sequence and (b) computational predictions. The dimensionless time period (g/d)1/21t between two consecutive images is ∼4.1.
between these two cases, yielding significant differences in their evolutions. Finally, the last situation we consider is typical of the large toroidal bubbles that develop when the Bond and Archimedes numbers are large enough.
Experimentally, the rise speed is obtained by tracking the successive positions of the bubble’s uppermost point with the image processing technique described in §2. Unless stated otherwise, we prefer to define the velocity V at this point rather than that of the bubble centroid because some parts of the surface may not be visible in experimental sequences involving dimpled bubbles, leading to errors in the determination of the centroid. In computations, spherical bubbles are released from rest with their centre located a distance 3d below the undisturbed fluid–fluid interface; this is why in the evolutions of the rise speed displayed below, this quantity is first seen to increase briefly, before reaching a broad maximum (see e.g. figures 5 and 7). Results are presented in dimensionless form using the bubble’s initial diameter d and the bubble volume V = πd3/6 to normalize positions and volumes, respectively, while velocities
and times are normalized by the gravitational scales (gd)1/2 and (d/g)1/2, respectively.
5.1. A small bubble trapped at the interface
We start by considering a small bubble with Bo = 3.0 and Ar = 7.65 in the upper series B (Λ = 1.11). As revealed by figure3(a), this corresponds to a case where, once the bubble has reached the region of the fluid–fluid interface, it remains trapped there throughout the rest of the period of observation because the film of heavy liquid that covers it has not yet been completely drained at the end of the sequence. Although the bubble is almost spherical before it reaches the interface region, its ‘final’ shape exhibits a marked top–bottom asymmetry with a fairly flat top region and a more rounded bottom part. Since the whole system is almost at rest, this is an indication that the hydrostatic pressure variations influence the local curvature of the bubble surface. Indeed, comparing the capillary length lc= (σ12/ρ1g)
1/2
with the bubble radius reveals that the ratio of both quantities is d/(2lc) = Bo1/2/2 ≈ 0.87.
According to figure 3, computational predictions differ dramatically from observations except during the early stages of the motion. The departure becomes visible in the third snapshot, when the computed bubble is no longer completely immersed in the lower fluid. Indeed the film that was still covering its top part in the previous snapshot has ruptured in the meantime. This is no surprise since the film was already very thin at that time (significantly less than d/10) and the local grid spacing is only d/100, implying that only a few grid cells lay in the gap. Then, for reasons
(a)
(b)
FIGURE 4. (Colour online) Evolution of the three-phase system for a bubble corresponding to Bo = 13.2 and Ar = 4.1 in the lower series B. Comparison between (a) the experimental sequence and (b) computational predictions. The dimensionless time period separating two consecutive images is ∼4.3.
similar to those discussed in §3 (see also appendix C), the flow in the gap is poorly described and numerical break-up soon occurs.
Although this computed evolution disagrees with the experimental observations, it deserves a few comments. As shown in figure 3, the computed bubble next spends a long time ‘floating’ on the interface (snapshots 3–6). A large meniscus first develops. Then it slowly recedes and a neck forms at the bottom of the bubble whose shape tends gradually toward that of a ‘hot air balloon’. This particular shape may be shown to be the only one that, given the values of I, S, R and Bo, satisfies both the Neumann condition expressing the equilibrium of the contact line (de Gennes et al. 2004) and the overall momentum balance expressing the vertical equilibrium of the bubble. In other words, if the bubble were released from rest right at the interface, it would take a shape and generate a meniscus very similar to those displayed in the last-but-one snapshot of the computational sequence in figure 3(b) and would stay there forever. However, in the present computation, the bubble is released well below the interface and the decrease in the potential energy of the whole system as it rises makes the kinetic energy of the fluid non-zero at the time it starts floating, even though some of this potential energy is dissipated by viscous effects and part of it is converted into interfacial energy through the increase of the bubble and fluid–fluid interface areas. This is why the floating configuration is only a transient in the present case and pinch-off finally takes place at the bottom of the bubble, entailing its release in the upper fluid.
5.2. Spheroidal bubbles
We now consider a situation belonging to the lower series B with a bubble nearly twice as big as that of the previous section. Hence the Bond number is Bo = 13.2 but the Archimedes number is only Ar = 4.1 owing to the high viscosity of the lower liquid. According to the experimental sequence displayed in figure 4(a), the bubble now succeeds in crossing the initial position of the fluid–fluid interface, although it is still covered by a film for some time. The film is seen to break up between the third and fourth snapshots, after which the bubble starts to become more elongated and to tow a column of heavy fluid. As the top of the column recedes along the bubble surface, the rear of the bubble becomes more pointed or even exhibits a small tip, owing to the stretching resulting from the combined effect of the bubble ascent and the recession of the column (snapshots 5–6). The column eventually separates from the bubble which then rises freely in the upper fluid.
0.3 0.2 0.1 –2 0 2 4 –2 0 2 4 4 3 2 1 – 4 6 – 4 6 8 0.4 0 5 0 (a) (b)
FIGURE 5. (Colour online) Evolution of (a) the normalized rise speed VT and (b) the displaced volume Ve as a function of the dimensionless bubble position zT for the situation considered in figure4:, experiment; solid line, computations. The error bars on VT result from the ±1 pixel uncertainty in the displacement of the uppermost point of the bubble between two successive images, while those onVe result from the 3-pixel uncertainty in the local radius of the entrained column.
The computed evolution correctly reproduces the various stages of the experimental sequence. In the late stages, the entrained column is noticeably thinner than its experimental counterpart at the same instant in time, but this is essentially because the computed bubble has travelled a somewhat larger distance (see below). Figure5(a) shows how the normalized rise speed VT = V/ (gd)1/2 evolves as a function of the
dimensionless position zT of the top of the bubble above the undisturbed position of
the interface. Starting from its steady value in the lower fluid, VT slightly reduces
when the top of the bubble approaches the fluid–fluid interface. Then it drops to less than half its initial value at the end of the period when the bubble is still covered by the film. After the film has ruptured, VT increases monotonically over a period of
time during which the bubble crosses a distance about twice its diameter, after which it almost reaches its new terminal value. The latter is somewhat less than it was in the lower fluid, essentially because of the reduction of the buoyancy force (R = 0.224). According to figure 5(a), the bubble Reynolds number Re = ArVT and Weber number
We = BoV2
T are of O(1) in both fluids. A theoretical prediction for the terminal velocity
taking into account inertial corrections corresponding to small Re and We was derived by Taylor & Acrivos (1964) using matched asymptotic expansions. It is of interest to examine how it compares with the experimental values of VT. Equating the buoyancy
force with the drag force predicted by Taylor & Acrivos in the case of a massless drop with zero internal viscosity yields the nonlinear equation
VT 1 +Re 8 + Re2 40 ln Re 2 + We 12 =Ar 12. (5.1)
With present values of Ar and Bo, (5.1) predicts VT≈0.278, which is in fairly good
agreement with, albeit slightly larger than, the initial value of VT (VT≈0.26) reported
in figure 5(a). The slight difference may be attributed to the contamination effects that tend to lower the rise speed. Note that if inertial corrections were neglected in
(5.1), one would predict VT≈0.342, which clearly overestimates the actual rise speed.
Replacing Ar by Aru and Bo by Bou in (5.1), the bubble terminal velocity in the upper
fluid is found to be VT≈0.230 which is in excellent agreement with the experimental
and computational values (contamination effects may decrease as the bubble rises in the upper fluid since silicon oils are known to be non-polar). The drop experienced by the rise speed when the bubble enters the region of the fluid–fluid interface is correctly predicted in the computation. However, for reasons already discussed above, the film that covers the top of the bubble breaks somewhat too early, preventing VT
from decreasing as much as it should and forcing it to start re-increasing slightly too early. The shift of the acceleration period toward lower positions could probably be resolved by a local increase of the grid resolution that would allow the film to subsist longer. However, we did not explore this possibility as we wish to determine how the computational approach deals with all physical situations under consideration with a single prescribed spatial resolution.
The volume of heavy fluid dragged into the upper fluid is a quantity of primary interest in three-phase systems. Depending on the context, it may for instance determine the mixing efficiency of the process or the amount of fluid that risks being projected if there is a free surface on top of the whole system (e.g. in iron processing where the upper layer is made of slag as discussed by Poggi et al. 1969, Reiter & Schwerdtfeger 1992a,b, Kobayashi 1993). We define this displaced volume as that of heavy fluid located above the position of the initial horizontal interface (i.e. we do not take into account the tiny reflux induced by mass conservation at large distance from the bubble path). Hereinafter this displaced volume is normalized by the bubble volume V , defining the dimensionless displaced volume Ve. As figure 5(b)
shows, Ve reaches a maximum ∼3 for zT ≈1, i.e. when the top of the bubble
is about one diameter above the undisturbed interface (third snapshot in figure 4). Then Ve decreases continuously until it reaches a constant small-but-non-zero value
(∼0.15) for zT > 5. This non-zero final value indicates that a small volume of lower
liquid remains permanently entrained by the bubble. Note that, counterintuitively, the comparison of figures 4 and 5(b) at various bubble positions indicates that the higher the liquid column entrained by the bubble, the smaller the total entrained volume. This is because the major contribution to the entrained volume comes from the region close to the fluid–fluid interface (where the surface of the displaced volume has a large radius) and not from the most visible part of the column (that attached to the rear of the bubble), whose radius is small.
We turn to a configuration belonging to series A with almost the same values of the Bond and Archimedes numbers (Bo = 13.3 and Ar = 4.15) as the one we just described; the main difference lies in the much lower viscosity of the upper fluid which, as will be seen, induces a markedly different evolution. As may be seen in figure 6(a), the first part of the sequence is similar to that observed in the previous case: after the bubble has started to deform the fluid–fluid interface, the film on its top part is quickly drained and the bubble starts to emerge in the upper fluid with a prolate shape. Then, as the vaguely hemispherical head of the bubble rises, a thin and long tail of air develops behind it, surrounded by a column of heavy fluid. This is because the low viscosity of the upper fluid allows the head of the bubble to rise fast while the part that is still in contact with the lower fluid is forced to rise much more slowly. After some time, the column of heavy fluid breaks right at the rear of the bubble head and starts receding. This in turn breaks the top of the bubble tail, part of which escapes from the entrained column and starts rising as an autonomous secondary bubble. The rest of the bubble tail recedes with the entrained column and eventually
(a)
(b)
FIGURE 6. (Colour online) Evolution of the three-phase system for a bubble corresponding to Bo = 13.3 and Ar = 4.15 in series A. Comparison between (a) the experimental sequence and (b) computational predictions. The dimensionless time period separating two consecutive images is ∼1.6. 0.6 0.2 0.4 4 3 2 1 5 0 (a) (b) –2 0 2 4 – 4 6 8 0.8 0 –2 0 2 4 – 4 6 8
FIGURE7. (Colour online) Evolution of (a) the normalized rise speed VTand (b) the displaced volumeVefor the bubble considered in figure6. See figure5for legend.
remains stuck just below the fluid–fluid interface. Note that after having released the column of heavy fluid, the main bubble undergoes significant shape oscillations.
The corresponding computational predictions are displayed in figure 6(b). The predicted shape and position of the main bubble are in good agreement with the experiments at each step of the sequence, although some differences may be noticed after the entrained column starts to form. In particular, the computations predict that a very thin column remains towed by the bubble at the end of the sequence although no such trend is detected experimentally. We also notice that the small rising bubble resulting from the break-up of the tail is not captured. Again, most of these secondary discrepancies could probably be removed by using a finer grid, but we did not explore this option.
Figure 7(a) shows how the normalized rise speed VT of the bubble evolves. Here
again, after starting from an initial steady-state value accurately predicted by (5.1), VT experiences some transient drop when the bubble starts to cross the fluid–fluid
interface, owing to the retarding effect of the film. Then VT increases by a factor of
which VT stabilizes itself at a value ∼0.6, is of course due to the large viscosity
contrast between the two liquids (Λ = 0.0175). However, the evolution of the bubble shape, which becomes very oblate (the bubble aspect ratio is ∼3.4 in the last snapshot of figure 6) and hence forces it to displace a large quantity of fluid as it rises, limits the increase of VT well below the value it would reach if the bubble had kept its
initial spherical shape (in which case the final rise speed would typically be 1/Λ larger than that in the lower fluid). According to the terminal value of VT at the end
of the sequence, the final bubble Weber number We = V2
TBou is ∼8.8 and the final
Reynolds number Re = VTAru is ∼110. Although the latter value might suggest that
the high-Re theory of Moore (1965) could be used to predict the terminal rise speed, this is actually not the case. The reason is that the bubble shape must be an oblate spheroid for this theory to hold, and such a shape exists in the high-Reynolds-number limit only if the Weber number is less than a critical value close to 3.23 (Miksis, Vanden-Broeck & Keller 1981; Meiron 1989). The present bubble is well beyond this limit, which explains why it oscillates while rising, and no theory is available to predict the rise speed under such conditions. According to figure 7(a), the computation correctly predicts the evolution of VT, although there is still some shift on the position
at which its sharp increase occurs. This shift is of course reminiscent of that observed in figure 5(a) and certainly has the same origin. During the final stage of the sequence, the predicted rise speed is seen to exhibit oscillations which are of course coupled to those of the bubble shape. The average value of VT in this final stage agrees well with
that deduced from experiments.
Figure 7(b) displays the evolution of the displaced volume. This volume first increases until it reaches a maximum about three times that of the bubble when zT≈2.
Then the entrained column starts receding, forcing Ve to decrease and eventually return
to zero for zT≈6. Note that the net buoyancy force acting on the bubble + column
system is positive only if Ve< (1 − R)/R ≈ 3. Hence, to sustain the bubble rise, the
normalized entrained volume cannot exceed (1 − R)/R, except during some transient stage. Figure 7(b) also shows that the computed evolution of the displaced volume is in close agreement with the experimental determination, although the retraction seems somewhat too slow, a direct consequence of the thin column that remains attached to the bubble in the late stages of the sequence.
5.3. Spherical cap bubbles
Let us now consider another bubble in series A with a diameter twice that of the previous one, i.e. Bo = 52.9 and Ar = 11.7. The corresponding sequence is displayed in figure 8. Here, since surface tension effects are weak, the rear part of the bubble is markedly dimpled when it reaches the fluid–fluid interface. After the short and thick tail it exhibits for some time has retracted, the bubble takes a final spherical cap shape with an angle close to 120◦. Given this shape, the column of liquid it entrains is much
thicker than in the previous case, although it gets thinner as the travelled distance increases and eventually breaks. The whole evolution is correctly predicted by the computations, including the transient presence of the aforementioned short thick tail (keep in mind that what is revealed by the experimental photographs is a side view of the bubble surface, while the computational snapshots show the bubble cross-section). The neck exhibited by the entrained column in the last experimental snapshot is also correctly captured.
Figure 9(a) shows how the normalized rise velocity VT of the top of the bubble
evolves. Remarkably, VT does not experience any jump (only a small bump) after the
(a)
(b)
FIGURE 8. (Colour online) Evolution of the three-phase system for a bubble corresponding to Bo = 52.9 and Ar = 11.7 in series A. Comparison between (a) the experimental sequence and (b) computational predictions. The dimensionless time period between two consecutive images is ∼1.5. 0.6 0.4 0.2 0.8 0 (a) (b) –1 0 1 2 3 4 – 2 5 – 2 –1 0 1 2 3 4 5 4 3 2 1 7 6 5 0
FIGURE9. (Colour online) Evolution of (a) the normalized rise speed VTand (b) the displaced volumeVefor the bubble considered in figure8. See figure5for legend.
figure 7(a). The reason for this difference may easily be identified: while the bubble considered in the previous case is spheroidal (at least until it has totally emerged in the upper fluid), the present one exhibits an almost spherical cap shape throughout its rise. Therefore the two rise speeds follow drastically different laws. In the former case, the drag force directly depends on the fluid viscosity (for a given bubble aspect ratio) because dissipation is essentially generated in the bulk of the fluid since the flow about the bubble is unseparated. In contrast, the flow past a spherical cap bubble is massively separated at the back of the bubble and most of the dissipation takes place there. Thus the flow in the front region is close to irrotational and, provided the Archimedes number (and hence the Reynolds number) is large enough, the rise velocity of a spherical cap bubble is known to depend only on its radius of curvature R∗ in that region through the relation V
T= (2/3)R1/2, where R = R∗/d (Davies &
Taylor 1950). This result can actually be extended to account for finite viscous effects, assuming that the flow is viscous but still irrotational in the front region, yielding
(a)
(b)
FIGURE10. (Colour online) Evolution of the three-phase system for a bubble corresponding
to Bo = 29.5 and Ar = 42.6 in the upper series B. Comparison between (a) the experimental sequence and (b) computational predictions. The dimensionless time period between two consecutive images is ∼1.7. (Joseph2003) VT= − 4 3ArR + 2 3R 1/2 1 + 4 Ar2R3 1/2 . (5.2)
Given the above prediction, there is no reason for VT to change abruptly when the
bubble emerges from the interface. Nevertheless, the secondary viscous corrections in (5.2) experience a jump due to the change in the value of the Archimedes number (from Ar to Aru). In the present case, this jump makes viscous corrections negligibly
small in the upper fluid, suggesting a small sudden increase of VT just after the
bubble has crossed the interface. This may be the origin of the small bump visible in figure 9(a). The mild increase of VT as the bubble rises in the upper liquid also
follows qualitatively the above prediction: as figure 8 shows, the bubble radius of curvature gradually increases (by ∼12 % between the last two snapshots), which in turn results in a gradual increase of VT. The rise speed is quantitatively well predicted
by the above formulae: in the last snapshot R ≈ 0.9, yielding VT ≈0.63 according to
(5.2), in close agreement with both experiment and computations. Not surprisingly, the displaced volume (figure 9b) reaches significantly larger values than in the previous case (figure 7b). The computations accurately capture the entire evolution of Ve,
although the small ‘plateau’ corresponding to the period during which the bubble crosses the interface seems somewhat exaggerated.
As in §5.2, it is of interest to examine how a large difference in the viscosity contrast Λ influences the evolution of bubbles belonging to the spherical cap family. For this purpose we consider a bubble with Bo = 29.5 and Ar = 42.6 in the upper series B. The corresponding evolution is displayed in figure 10. Since this bubble has a smaller Bond number compared to that of series A discussed above, it reaches the interface with a less dimpled shape and results in a spherical cap of smaller radius in the upper fluid. However, the similarity between the two shape evolutions is clear and is confirmed by the resemblances between the evolutions of the two rise speeds (figures9aand 11a).
The fact that the actual Archimedes number only decreases by ∼30 % in the present case and remains moderate after the bubble has crossed the interface (Aru ≈30),
whereas it roughly increases by a factor of fifty in the previous case and becomes large (Aru≈500), has two main consequences. First, the column of entrained liquid
0.6 0.4 0.2 1.0 0.8 0 (a) (b) –1 0 1 2 3 4 – 2 5 6 –1 0 1 2 3 4 – 2 5 6 4 2 10 8 6 0
FIGURE11. (Colour online) Evolution of (a) the normalized rise speed VTand (b) displaced volumeVefor the bubble considered in figure10. See figure5for legend.
is somewhat thicker here (compare the end of the sequences in figures 8 and 10), yielding a maximum displaced volume (figure 11b) ∼50 % larger. Second, the time required for the bubble to reach a stationary shape in the upper fluid is much shorter when Aru is moderate. This is why the rise velocity in figure 11(a) has already
reached a steady value at the end of the sequence, while that in figure 9(a) is still increasing. The viscosity jump also influences the rise speed of spherical cap bubbles in an indirect and subtle manner. Indeed, the leading-order relation VT∝ R1/2
implies V2
TS ∝ R
3, where S denotes the area of the bubble’s horizontal
cross-section. Therefore the balance of drag and buoyancy forces implies that, throughout the bubble ascent in each fluid, the variations of the bubble drag coefficient CD and
those of its radius or curvature are linked by the condition CDR3≈const. (there
may be some variation of the cap angle, so S /R2 may not be strictly constant).
The drag coefficient of these bubbles is a decreasing function of the Archimedes number, being essentially determined by the viscous dissipation in the wake. Therefore when Ar jumps to Aru and the above constant jumps from 1 to 1 − R after the
bubble has crossed the interface, R has to increase or decrease, depending on whether (1 − R)C−1D (Aru) is larger or smaller than C−1D (Ar), and the variation of VT follows.
This implies that R and VT have to increase when R is small and Λ ≪ 1, while the
density contrast forces them to decrease slightly when Λ ≈ 1. These predictions may be verified in figures 8and 9(respectively figures 10 and11), which correspond to the first (respectively second) scenario.
Note that, compared to the bubble of the lower series B examined in §5.2 (whose diameter is about two-thirds that of the present bubble), no sharp drop of the rise velocity is observed in figure 11 when the bubble approaches the horizontal interface. This is an indication that the capillary overpressure in the film that covers the bubble during this stage barely influences its dynamics, as will be discussed in more detail in the next section. The only discernible effect of this film is the small bump visible in the evolution of VT just before it reaches its final value: the corresponding video
sequence indicates that this bump is associated with the late rupture of the film; a similar, albeit weaker and somewhat premature bump may also be discerned in the computations. The experimental/computational rise speed at the end of the sequence
(a) (b)
(c) (d )
FIGURE 12. (Colour online) Flow pattern past the bubble considered in figure 10 at the time when its top stands at zT≈3.4. (a,b) Streamline and velocity fields obtained through PIV measurements; (c,d) streamline and vorticity fields taken from computations. The azimuthal vorticity ωθ is normalized by VT(g/d)1/2. The red (respectively black/blue) regions correspond to positive (respectively negative) vorticity.
compares well with the inviscid prediction VT = (2/3)R1/2 since the latter predicts
VT ≈0.58 (R ≈ 0.75). Interestingly, the agreement with the prediction of (5.2) is not
as good (VT ≈0.52), which may suggest that the assumption of viscous potential
flow overestimates the influence of viscous effects on the rise speed. This is known to be the case with spherical bubbles for which the boundary layer resulting from the non-zero surface vorticity lowers the drag force and hence increases the rise speed (Moore 1963). Although the present result is obviously not sufficient to settle the matter definitely, it suggests that taking into account the normal viscous stresses (from which the viscous correction in (5.2) arises) without considering the shear-free condition responsible for the surface vorticity (which modifies the pressure distribution along the interface) may not be a suitable approximation.
Figure 12 displays PIV and computational determinations of the flow pattern around the bubble at the time when the entrained volume reaches its maximum. Not surprisingly, the upper half of the streamline pattern is dominated by a dipole structure associated with the bubble motion. While the upper part of the entrained column is still rising with the bubble, its lower part is already receding towards its initial position. The central part of the column is thus stretched axially, which creates the hyperbolic point visible on the column axis. At the same time the base of the column thickens and flows radially on the horizontal interface, which induces an axial compression and results in another hyperbolic point near the intersection of
the column axis with the plane of the undisturbed interface. Positive values of the azimuthal vorticity ωθ are of course concentrated close to the bubble surface, with a
dimensionless maximum ωmax≈23.1 located slightly ahead of the region of maximum curvature where most of the vorticity is produced. A small, nearly horizontal and almost black zone can be discerned just at the back of the bubble, in the region where its surface is concave. This corresponds to negative vorticity (with a minimum ωmin≈ −7.2) generated in the attached eddy, where the tangential velocity goes from the bubble axis to the region of maximum curvature. Finally a thin shear layer having negative vorticity (with ωmin≈ −1.6) is observed along the fluid–fluid interface in the lower part of the column; this structure is of course a direct consequence of the entrainment of the outer fluid, which has low downward velocities, by the heavy fluid which recedes faster.
5.4. A toroidal bubble
We finally consider the situation corresponding to Bo = 48.2 and Ar = 7840 in series C, which, according to our observations, results in a toroidal bubble. Indeed, when viscous effects are small enough, large initially nearly spherical bubbles are known to undergo a topological change and become toroidal. The basic mechanism that drives this transition is the growth of the tongue that forms at the bottom of the bubble, owing to the hydrostatic pressure difference between the top and bottom regions. As it develops, this tongue dramatically reduces the height of the bubble along its axis and quickly leads to pinch-off unless capillary forces are strong enough to limit its development. Bonometti & Magnaudet (2006) showed that, when Ar > 103, this
topological transition always happens with initially spherical bubbles whose Bond number is beyond a critical value in the range 32–35. Therefore the case considered here is supercritical and the bubble switches from its initial shape to the toroidal configuration well before it reaches the fluid–fluid interface. Toroidal bubbles rising in low-viscosity fluids have been studied experimentally (Walters & Davidson 1963) and theoretically (Pedley 1968), the bubble then being considered as a hollow vortex ring. Given the large values of Ar generally associated with such bubbles, their dynamics are essentially inviscid during most of their lifetime. Provided the core has a circular cross-section, the evolution of the dimensionless ring radius R(t) and rise velocity V(t) is then governed by R(t) R0 = 1 + t − t0 6Γ R2 0 1/2 , V(t) = Γ 4πR log 16 (3πR3)1/2−1 2 , (5.3)
where R0= R(t0) and the circulation Γ around the core has been normalized by
(gd3)1/2. The core radius a(t) may then be determined thanks to the condition of
volume conservation 12πR(t)a2(t) =1. Toroidal bubbles rising in inviscid fluid have
been investigated computationally by Lundgren & Mansour (1991) using a BIM, well before the full Navier–Stokes computations of Bonometti & Magnaudet (2006). Their results confirmed inviscid predictions and revealed the existence of significant oscillations of the ring radius and core shape when the ratio of the ring-to-core radii is small enough.
In the present context, as figure 13 shows, the most salient feature revealed by experiments as well as by computations is that the bubble remains encapsulated in a thick body of heavy fluid after having crossed the original position of the interface. This is due to the circulation around the bubble core which induces an upward fluid velocity in the central part of the ring (as may be discerned in figure 14), which in
(a)
(b)
(c)
FIGURE13. (Colour online) Evolution of the three-phase system for a bubble corresponding to Bo = 48.2 and Ar = 7840 in series C. (a) Experimental sequence and (b,c) computational predictions. The dimensionless time period between two consecutive images is ∼0.66 in the upper two rows and ∼1.25 in the bottom row.
FIGURE14. (Colour online) Experimental velocity and streamline patterns past a 6 cm3(d ≈ 22.5 mm) toroidal bubble just after it has crossed the position of the undisturbed interface.
turn tends to ‘feed’ the film ahead of the bubble with fresh heavy fluid withdrawn from the entrained column. Note that, according to the computational sequences, the core is far from circular and undergoes strong oscillations, in line with the findings of Lundgren and Mansour.
Unfortunately, comparison of present experimental results with theoretical and computational predictions for R(t) and V(t) can hardly be made quantitative, for several reasons. First, it must be kept in mind that the original bubble is injected much closer to the undisturbed interface in the computation than in the experiment. Moreover, given the design of the injection system, this original bubble is certainly not spherical in the latter, and this has a direct influence on the time (i.e. height) at which the topological transition takes place as well as on the value of the circulation. Indeed Bonometti & Magnaudet (2006) showed that under certain conditions, bubbles with
1.2 0.8 0.4 0 0 2 0 2 1.2 0.8 0.4 0 2 8 4 –1 0 1 2 –1 0 1 2 1.6 0 R V 12 0 –2 4 –2 4 –2 4 (a) –2 3 (b) (c) tc –2 3 tc
FIGURE 15. (Colour online) Evolution of (a) the normalized ring radius R, (b) rise speed V, and (c) entrained volume Ve for the toroidal bubble displayed in figure 13 versus the dimensionless vertical position zc of the bubble’s centre of inertia above the undisturbed interface (black squares and solid line). Subfigures (a) and (b) also display the corresponding evolutions versus the dimensionless time tc, with tc=0 when zc=0 (open circles and dotted line). The ‘experimental’ ring and core radii are obtained by evaluating the dimensionless area S∗ and perimeter P∗ of the projected surface S of the bubble on images such as that of figure 14, and equating these quantities with their counterparts for a torus with a circular core, which yields the conditions S∗=4Ra + πa2 and P∗ =4R + 2πa; the rise velocity is defined as the time variation of the vertical position of the centre of inertia of S and the entrained volume is obtained by fitting the fluid–fluid interface contour with a curve preserving the left–right symmetry, evaluating the enclosed volume by invoking axisymmetry and removing the known bubble volume; this procedure does not work when the bubble crosses the undisturbed position of the interface, which is why no data appear in (c) during that stage. The computational ring radius is defined as R(t) =R r2C(r, z, t)dr dz /R rC(r, z, t) dr dz, where C is the local volume fraction of air and integration is performed over the whole computational domain; the rise velocity is defined as V(t) =R rUz(r, z, t)C(r, z, t)dr dz /R rC(r, z, t) dr dz, where Uz is the local vertical fluid velocity.
a large enough initial oblateness can even preserve a spherical cap shape throughout their life, whereas spherical bubbles with the same volume quickly become toroidal. Because of these two factors, there is no chance that real and computed bubbles of a given volume give birth to a toroidal bubble at the same distance from the interface. Therefore, the corresponding two toroidal bubbles do not have the same ‘age’ when they reach the position of the undisturbed interface, nor do they probably have the same initial ring radius R0 and circulation Γ . Also, given the large value
of the Archimedes number, the actual bubble shape and the flow about it are three-dimensional (with significant azimuthal fluctuations as may be seen in figure 14), whereas available predictions and present computations assume an axisymmetric evolution.
Having pointed out these issues, we are left with the possibility of performing qualitative comparisons. Figure 15 displays the evolution of the ring radius R, rise velocity V and entrained volume Ve; the way these quantities are defined and extracted
from the complex bubble and interface shapes revealed by figure 13 is detailed in the corresponding caption. Figure 15(b) shows that the rise velocity is a slowly decreasing function of time and vertical position, a trend that was to be expected owing to the R−1log R term in the second of (5.3). The experimental and computational evolutions
